Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,9,11,12)(15,19)(17,20)(21,22)(23,24)(25,33)(26,27)(28,31)(29,34)(30,35)(32,36);(2,4)(13,14,16,18)(15,17,19,20)(22,27)(24,28)(29,33)(32,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,33)(34,36);(5,7,8,11)(6,9,10,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,28)(22,23,27,31)(25,32,34,35)(29,30,33,36);(21,25,26,34)(22,29,27,33)(23,30,31,36)(24,32,28,35)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 526888);